Let’s call a group $\frac{3}{2}$-generated iff for every $g \in G \setminus \{1\}$ there is some $h \in G \setminus \langle g \rangle$ with $G = \langle g,h \rangle$.
There is a conjecture by Breuer, Guralnick and Kantor, that a finite group is $\frac{3}{2}$-generated iff all its proper quotients are cyclic.
Why is the word “finite” in the conjecture necessary? Is there an infinite counterexample? If yes, what it is? A $\frac{3}{2}$-generated group with a non-cyclic proper factor? Or a non-$\frac{3}{2}$-generated group, all proper cycles of which are cyclic?
Let $G$ be any simple group that is not finitely generated (for instance, the alternating group of finite-support permutations of an infinite set). Then clearly $G$ is not $3/2$-generated, but every proper quotient is trivial and in particular cyclic.